2023 Cayley Problem 24

Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is \frac{1}{2}. After she wins a game, the probability of Carina winning the next game is \frac{3}{4}. After she loses a game, the probability of Carina winning the next game is \frac{1}{3}. The probability that Carina wins 3 games before being eliminated from the tournament equals \frac{a}{b}, where the fraction \frac{a}{b} is in lowest terms. What is the value of a + b?

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.